An inequality is given for the expected length of a confidence interval given that a particular distribution generated the data and assuming that the confidence interval has a given coverage probability over a family of distributions. As a corollary, attempts to adapt to the regularity of the true density within derivative smoothness classes cannot improve the rate of convergence of the length of the confidence interval over minimax fixed-length intervals and still maintain uniform coverage probability. However, adaptive confidence intervals can attain improved rates of convergence in some other classes of densities, such as those satisfying a shape restriction.
"On nonparametric confidence intervals." Ann. Statist. 25 (6) 2547 - 2554, December 1997. https://doi.org/10.1214/aos/1030741084